Traveling Waves in a One-Dimensional Heterogeneous Medium

Nolen, James; Ryzhik, Lenya

doi:10.1016/j.anihpc.2009.02.003

Nolen, James ; Ryzhik, Lenya

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 1021-1047.

MR Zbl | 3 citations dans Numdam

DOI : 10.1016/j.anihpc.2009.02.003

@article{AIHPC_2009__26_3_1021_0,
     author = {Nolen, James and Ryzhik, Lenya},
     title = {Traveling {Waves} in a {One-Dimensional} {Heterogeneous} {Medium}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1021--1047},
     publisher = {Elsevier},
     volume = {26},
     number = {3},
     year = {2009},
     doi = {10.1016/j.anihpc.2009.02.003},
     mrnumber = {2526414},
     zbl = {1178.35205},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.02.003/}
}

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Nolen, James; Ryzhik, Lenya. Traveling Waves in a One-Dimensional Heterogeneous Medium. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 1021-1047. doi : 10.1016/j.anihpc.2009.02.003. http://www.numdam.org/articles/10.1016/j.anihpc.2009.02.003/

Bibliographie
Cité par

[1] Angenent S. B., The Zero Set of a Solution of a Parabolic Equation, J. Reine Angew. Math. 390 (1988) 79-96. | MR | Zbl

[2] Berestycki H., Hamel F., Fronts in Periodic Excitable Media, Comm. Pure Appl. Math. 60 (2002) 949-1032. | MR | Zbl

[3] Berestycki H., Hamel F., Fronts and Invasions in General Domains, C. R. Acad. Sci. Paris Ser. I 343 (2006) 711-716. | MR | Zbl

[4] Berestycki H., Hamel F., Generalized Travelling Waves for Reaction-Diffusion Equations, in: Perspectives in Nonlinear Partial Differential Equations. in Honor of H. Brezis, Contemp. Math., vol. 46, Amer. Math. Soc., 2007, pp. 101-123. | MR

[5] Berestycki H., Nirenberg L., On the Method of Moving Planes and the Sliding Method, Bull. Braz. Math. Soc. (N.S.) 22 (1991) 1-37. | MR | Zbl

[6] Fife P. C., Mcleod J. B., The Approach of Solutions of Nonlinear Diffusion Equations by Traveling Front Solutions, Arch. Ration. Mech. Anal. 65 (1977) 335-361. | MR | Zbl

[7] Freidlin M., Gärtner J., On the Propagation of Concentration Waves in Periodic and Random Media, Soviet Math. Dokl. 20 (1979) 1282-1286. | MR | Zbl

[8] Freidlin M., Functional Integration and Partial Differential Equations, Ann. of Math. Stud., vol. 109, Princeton Univ. Press, Princeton, NJ, 1985. | MR | Zbl

[9] Freidlin M., Limit Theorems for Large Deviations and Reaction-Diffusion Equations, Ann. Probab. 13 (1985) 639-675. | MR | Zbl

[10] Freidlin M., Geometric Optics Approach to Reaction-Diffusion Equations, SIAM J. Appl. Math. 46 (1986) 222-232. | MR | Zbl

[11] Freidlin M., Reaction-Diffusion in Incompressible Fluid: Asymptotic Problems, J. Differential Equations 179 (2002) 44-96. | MR | Zbl

[12] Hamel F., Nadirashvili N., Entire Solutions of the KPP Equation, Comm. Pure Appl. Math. 52 (1999) 1255-1276. | MR | Zbl

[13] Hamel F., Nadirashvili N., Travelling Fronts and Entire Solutions of the Fisher-KPP Equation in $R^{N}$ , Arch. Ration. Mech. Anal. 157 (2001) 91-163. | MR | Zbl

[14] Kanel Ya., Stabilization of Solutions of the Cauchy Problem for Equations Encountered in Combustion Theory, Mat. Sb. 59 (1962) 245-288. | MR

[15] Kosygina E., Rezakhanlou F., Varadhan S. R.S., Stochastic Homogenization of Hamilton-Jacobi-Bellman Equations, Comm. Pure Appl. Math. 59 (2006) 1489-1521. | MR | Zbl

[16] Krylov N. V., Safonov M. V., A Property of the Solutions of Parabolic Equations With Measureable Coefficients, (English translation), Izv. Akad. Nauk SSSR Ser. Mat. 16 (1) (1981) 151-164. | MR | Zbl

[17] Lewis T. J., Keener J. P., Wave-Block in Excitable Media Due to Regions of Depressed Excitability, SIAM J. Appl. Math. 61 (2000) 293-316. | MR | Zbl

[18] Liggett T. M., An Improved Subadditive Ergodic Theorem, Ann. Probab. 13 (1985) 1279-1285. | Zbl

[19] Lions P.-L., Souganidis P., Homogenization of “viscous” Hamilton-Jacobi Equations in Stationary Ergodic Media, Comm. Partial Differential Equations 30 (2005) 335-375. | Zbl

[20] Majda A., Souganidis P., Large Scale Front Dynamics for Turbulent Reaction-Diffusion Equations With Separated Velocity Scales, Nonlinearity 7 (1994) 1-30. | Zbl

[21] Mallordy J.-F., Roquejoffre J.-M., A Parabolic Equation of the KPP Type in Higher Dimensions, SIAM J. Math. Anal. 26 (1995) 1-20. | Zbl

[22] H. Matano, talks presented at various conferences.

[23] A. Mellet, J.-M. Roquejoffre, Y. Sire, Generalized fronts for one-dimensional reaction-diffusion equations, preprint, 2008. | MR | Zbl

[24] Nolen J., Xin J., Variational Principle of KPP Front Speeds in Temporally Random Shear Flows With Applications, Comm. Math. Phys. 269 (2007) 493-532. | MR | Zbl

[25] J. Nolen, J. Xin, Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows, Ann. Inst. H. Poincaré Anal. Non Linéaire (2008), in press, 10.1016/j.anihpc.2008.02.005. | Numdam | MR | Zbl

[26] Roquejoffre J.-M., Eventual Monotonicity and Convergence to Traveling Fronts for the Solutions of Parabolic Equations in Cylinders, Ann. Inst. H. Poincaré 14 (4) (1997) 499-552. | Numdam | MR | Zbl

[27] Shen W., Traveling Waves in Diffusive Random Media, J. Dynam. Differential Equations 16 (4) (2004) 1011-1060. | MR | Zbl

[28] Weinberger H., On Spreading Speeds and Traveling Waves for Growth and Migration Models in a Periodic Habitat, J. Math. Biol. 45 (2002) 511-548. | MR | Zbl

[29] Xin J., Existence of Planar Flame Fronts in Convective-Diffusive Periodic Media, Arch. Ration. Mech. Anal. 121 (1992) 205-233. | MR | Zbl

[30] Xin J., Existence and Nonexistence of Traveling Waves and Reaction-Diffusion Front Propagation in Periodic Media, J. Stat. Phys. 73 (1993) 893-926. | MR | Zbl

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